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September  2021, 41(9): 4085-4104. doi: 10.3934/dcds.2021029

## Best approximation of orbits in iterated function systems

 1 School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan 430074, China 2 School of Science, Wuhan University of Technology, Wuhan 430074, China

* Corresponding author: Bo Tan

Received  October 2020 Revised  December 2020 Published  September 2021 Early access  January 2021

Let
 $\Phi = \{\phi_{i}\colon i\in\Lambda\}$
be an iterated function system on a compact metric space
 $(X,d)$
, where the index set
 $\Lambda = \{1, 2, \ldots,l\}$
with
 $l \ge2$
, or
 $\Lambda = \{1,2,\ldots\}$
. We denote by
 $J$
the attractor of
 $\Phi$
, and by
 $D$
the subset of points possessing multiple codings. For any
 $x\in J\backslash D,$
there is a unique integer sequence
 $\{\omega_{n}(x)\}_{n\geq 1}\subset \Lambda^{\mathbb{N}}$
, called the digit sequence of
 $x,$
such that
 $\{x\} = \bigcap\limits_n\phi_{\omega_{1}(x)}\circ\cdots\circ\phi_{\omega_{n}(x)}(X).$
In this case we write
 $x = [\omega_{1}(x),\omega_{2}(x),\ldots].$
For
 $x, y\in J\backslash D,$
we define the shortest distance function
 $M_{n}(x,y)$
as
 \begin{align*} M_{n}(x,y) = \max\big\{k\in \mathbb{N}\colon \omega_{i+1}(x) = \omega_{i+1}(y),\ldots, &\omega_{i+k}(x) = \omega_{i+k}(y) \; \\&\text{for some}\; 0\leq i \leq n-k\big\}, \end{align*}
which counts the run length of the longest same block among the first
 $n$
digits of
 $(x,y).$
In this paper, we are concerned with the asymptotic behaviour of
 $M_{n}(x,y)$
as
 $n$
tends to
 $\infty.$
We calculate the Hausdorff dimensions of the exceptional sets arising from the shortest distance function. As applications, we study the exceptional sets in several concrete systems such as continued fractions system, Lüroth system,
 $N$
-ary system, and triadic Cantor system.
Citation: Saisai Shi, Bo Tan, Qinglong Zhou. Best approximation of orbits in iterated function systems. Discrete and Continuous Dynamical Systems, 2021, 41 (9) : 4085-4104. doi: 10.3934/dcds.2021029
##### References:
 [1] M. D. Boshernitzan, Quantitative recurrence results, Invent. Math., 113 (1993), 617-631.  doi: 10.1007/bf01244320. [2] D. Bessis, G. Paladin, G. Turchetti and S. Vaienti, Generalized dimensions, entropies, and Liapunov exponents from the pressure function for strange sets, J. Statist. Phys., 51 (1988), 109-134.  doi: 10.1007/bf01015323. [3] L. Barreira and B. Saussol, Hausdorff dimension of measures via Poincaré recurrence, Comm. Math. Phys., 219 (2001), 443-463.  doi: 10.1007/s002200100427. [4] Y. Bugeaud and B.-W. Wang, Distribution of full cylinders and the Diophantine properties of the orbits in $\beta$-expansions, J. Fractal Geom., 1 (2014), 221-241.  doi: 10.4171/jfg/6. [5] I. P. Coornfeld, S. V. Fomin and Ya. G. Sinaǐ, Ergodic Theory, Springer-Verlag, New York, 1982. [6] N. Chernov and D. Kleinbock, Dynamical Borel-Cantelli lemmas for Gibbs measures, Israel J. Math., 122 (2001), 1-27.  doi: 10.1007/bf02809888. [7] K. Falconer, Fractal Geometry, Mathematical Foundations and Applications, 3$^nd$ edition, John Wiley & Sons, Ltd., Chichester, 2014. [8] J. L. Fernández, M. V. Melián and D. Pestana, Quantitative recurrence properties of expanding maps, preprint, arXiv: math/0703222. [9] S. Galatolo, Dimension and hitting time in rapidly mixing systems, Math. Res. Lett., 14 (2007), 797-805.  doi: 10.4310/mrl.2007.v14.n5.a8. [10] G. H. Hardy and E. M. Wright, An Introduction to The Theory of Numbers, 5$^nd$ edition, The Clarendon Press, Oxford University Press, New York, 1979. [11] R. Hill and S. L. Velani, The ergodic theory of shrinking targets, Invent. Math., 119 (1995), 175-198.  doi: 10.1007/bf01245179. [12] R. Hill and S. L. Velani, Metric Diophantine approximation in Julia sets of expanding rational maps, Inst. Hautes Études Sci. Publ. Math., (1997), 193–216. doi: 10.1007/bf02699537. [13] N. Haydn and S. L. Vaienti, The Rényi entropy function and the large deviation of short return times, Ergodic Theory Dynam. Systems., 30 (2010), 159-179.  doi: 10.1017/s0143385709000030. [14] H. Jager and C. de Vroedt, Lüroth series and their ergodic properties, Nederl. Akad. Wetensch. Proc. Ser., 31 (1969), 31-42.  doi: 10.1016/1385-7258(69)90023-7. [15] A. Ya. Khintchine, Continued Fractions, Translated by Peter WynnP. Noordhoff, Ltd., Groningen 1963. doi: 10.1017/s0008439500032033. [16] B. Li, B. W. Wang, J. Wu and J. Xu, The shrinking target problem in the dynamical system of continued fractions, Proc. London Math. Soc., 108 (2014), 159-186.  doi: 10.1112/plms/pdt017. [17] J. Li and X. Yang, On longest matching consecutive subsequence, Int. J. Number Theory., 15 (2019), 1745-1758.  doi: 10.1142/s1793042119500970. [18] R. D. Mauldin and M. Urbański, Dimensions and measures in infinite iterated function systems, Proc. London Math. Soc., 73 (1996), 105-154.  doi: 10.1112/plms/s3-73.1.105. [19] R. D. Mauldin and M. Urbański, Conformal iterated function systems with applications to the geometry of continued fractions, Trans. Amer. Math. Soc., 351 (1999), 4995-5025.  doi: 10.1090/s0002-9947-99-02268-0. [20] D. S. Ornstein and B. Weiss, Entropy and data compression schemes, IEEE Trans. Inform. Theory., 39 (1993), 78-83.  doi: 10.1109/18.179344. [21] L. Peng, On the hitting depth in the dynamical system of continued fractions, Chaos. Solitons. Fractals., 69 (2014), 22-30.  doi: 10.1016/j.chaos.2014.09.003. [22] L. Peng, B. Tan and B. W. Wang, Quantitative Poincaré recurrence in continued fraction dynamical system, Sci. China Math., 55 (2012), 131-140.  doi: 10.1007/s11425-011-4303-9. [23] B. Saussol, Recurrence rate in rapidly mixing dynamical systems, Discrete Contin. Dyn. Syst., 15 (2006), 259-267.  doi: 10.3934/dcds.2006.15.259. [24] B. O. Stratmann and M. Urbánski, Jarník, Julia$\colon$a Diophantine analysis for geometrically finite Kleinian groups with parabolic elements, Math. Scand., 91 (2002), 27-54.  doi: 10.7146/math.scand.a-14377. [25] S. Seuret and B.-W. Wang, Quantitative recurrence properties in conformal iterated function systems, Adv. Math., 280 (2015), 472-505.  doi: 10.1016/j.aim.2015.02.019. [26] B. Tan and B.-W. Wang, Quantitative recurrence properties for beta-dynamical system, Adv. Math., 228 (2011), 2071-2097.  doi: 10.1016/j.aim.2011.06.034. [27] F. Takens and E. Verbitski, Generalized entropies$\colon$Rényi and correlation integral approach, Nonlinearity., 11 (1998), 771-782.  doi: 10.1088/0951-7715/11/4/001. [28] B. Tan and Q. L. Zhou, The relative growth rate for partial quotients in continued fractions, J. Math. Anal. Appl., 478 (2019), 229-235.  doi: 10.1016/j.jmaa.2019.05.029. [29] P. Walters, An Introduction to Ergodic Theory, Graduate Texts in Mathematics, 79. Springer-Verlag, New York-Berlin, 1982. [30] Q.-L. Zhou, Dimensions of recurrent sets in $\beta$-symbolic dynamics, J. Math. Anal. Appl., 472 (2019), 1762-1776.  doi: 10.1016/j.jmaa.2018.12.022.

show all references

##### References:
 [1] M. D. Boshernitzan, Quantitative recurrence results, Invent. Math., 113 (1993), 617-631.  doi: 10.1007/bf01244320. [2] D. Bessis, G. Paladin, G. Turchetti and S. Vaienti, Generalized dimensions, entropies, and Liapunov exponents from the pressure function for strange sets, J. Statist. Phys., 51 (1988), 109-134.  doi: 10.1007/bf01015323. [3] L. Barreira and B. Saussol, Hausdorff dimension of measures via Poincaré recurrence, Comm. Math. Phys., 219 (2001), 443-463.  doi: 10.1007/s002200100427. [4] Y. Bugeaud and B.-W. Wang, Distribution of full cylinders and the Diophantine properties of the orbits in $\beta$-expansions, J. Fractal Geom., 1 (2014), 221-241.  doi: 10.4171/jfg/6. [5] I. P. Coornfeld, S. V. Fomin and Ya. G. Sinaǐ, Ergodic Theory, Springer-Verlag, New York, 1982. [6] N. Chernov and D. Kleinbock, Dynamical Borel-Cantelli lemmas for Gibbs measures, Israel J. Math., 122 (2001), 1-27.  doi: 10.1007/bf02809888. [7] K. Falconer, Fractal Geometry, Mathematical Foundations and Applications, 3$^nd$ edition, John Wiley & Sons, Ltd., Chichester, 2014. [8] J. L. Fernández, M. V. Melián and D. Pestana, Quantitative recurrence properties of expanding maps, preprint, arXiv: math/0703222. [9] S. Galatolo, Dimension and hitting time in rapidly mixing systems, Math. Res. Lett., 14 (2007), 797-805.  doi: 10.4310/mrl.2007.v14.n5.a8. [10] G. H. Hardy and E. M. Wright, An Introduction to The Theory of Numbers, 5$^nd$ edition, The Clarendon Press, Oxford University Press, New York, 1979. [11] R. Hill and S. L. Velani, The ergodic theory of shrinking targets, Invent. Math., 119 (1995), 175-198.  doi: 10.1007/bf01245179. [12] R. Hill and S. L. Velani, Metric Diophantine approximation in Julia sets of expanding rational maps, Inst. Hautes Études Sci. Publ. Math., (1997), 193–216. doi: 10.1007/bf02699537. [13] N. Haydn and S. L. Vaienti, The Rényi entropy function and the large deviation of short return times, Ergodic Theory Dynam. Systems., 30 (2010), 159-179.  doi: 10.1017/s0143385709000030. [14] H. Jager and C. de Vroedt, Lüroth series and their ergodic properties, Nederl. Akad. Wetensch. Proc. Ser., 31 (1969), 31-42.  doi: 10.1016/1385-7258(69)90023-7. [15] A. Ya. Khintchine, Continued Fractions, Translated by Peter WynnP. Noordhoff, Ltd., Groningen 1963. doi: 10.1017/s0008439500032033. [16] B. Li, B. W. Wang, J. Wu and J. Xu, The shrinking target problem in the dynamical system of continued fractions, Proc. London Math. Soc., 108 (2014), 159-186.  doi: 10.1112/plms/pdt017. [17] J. Li and X. Yang, On longest matching consecutive subsequence, Int. J. Number Theory., 15 (2019), 1745-1758.  doi: 10.1142/s1793042119500970. [18] R. D. Mauldin and M. Urbański, Dimensions and measures in infinite iterated function systems, Proc. London Math. Soc., 73 (1996), 105-154.  doi: 10.1112/plms/s3-73.1.105. [19] R. D. Mauldin and M. Urbański, Conformal iterated function systems with applications to the geometry of continued fractions, Trans. Amer. Math. Soc., 351 (1999), 4995-5025.  doi: 10.1090/s0002-9947-99-02268-0. [20] D. S. Ornstein and B. Weiss, Entropy and data compression schemes, IEEE Trans. Inform. Theory., 39 (1993), 78-83.  doi: 10.1109/18.179344. [21] L. Peng, On the hitting depth in the dynamical system of continued fractions, Chaos. Solitons. Fractals., 69 (2014), 22-30.  doi: 10.1016/j.chaos.2014.09.003. [22] L. Peng, B. Tan and B. W. Wang, Quantitative Poincaré recurrence in continued fraction dynamical system, Sci. China Math., 55 (2012), 131-140.  doi: 10.1007/s11425-011-4303-9. [23] B. Saussol, Recurrence rate in rapidly mixing dynamical systems, Discrete Contin. Dyn. Syst., 15 (2006), 259-267.  doi: 10.3934/dcds.2006.15.259. [24] B. O. Stratmann and M. Urbánski, Jarník, Julia$\colon$a Diophantine analysis for geometrically finite Kleinian groups with parabolic elements, Math. Scand., 91 (2002), 27-54.  doi: 10.7146/math.scand.a-14377. [25] S. Seuret and B.-W. Wang, Quantitative recurrence properties in conformal iterated function systems, Adv. Math., 280 (2015), 472-505.  doi: 10.1016/j.aim.2015.02.019. [26] B. Tan and B.-W. Wang, Quantitative recurrence properties for beta-dynamical system, Adv. Math., 228 (2011), 2071-2097.  doi: 10.1016/j.aim.2011.06.034. [27] F. Takens and E. Verbitski, Generalized entropies$\colon$Rényi and correlation integral approach, Nonlinearity., 11 (1998), 771-782.  doi: 10.1088/0951-7715/11/4/001. [28] B. Tan and Q. L. Zhou, The relative growth rate for partial quotients in continued fractions, J. Math. Anal. Appl., 478 (2019), 229-235.  doi: 10.1016/j.jmaa.2019.05.029. [29] P. Walters, An Introduction to Ergodic Theory, Graduate Texts in Mathematics, 79. Springer-Verlag, New York-Berlin, 1982. [30] Q.-L. Zhou, Dimensions of recurrent sets in $\beta$-symbolic dynamics, J. Math. Anal. Appl., 472 (2019), 1762-1776.  doi: 10.1016/j.jmaa.2018.12.022.
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